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  <div class="section" id="numpy-i0">
<h1>numpy.i0<a class="headerlink" href="#numpy-i0" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="numpy.i0">
<code class="sig-prename descclassname">numpy.</code><code class="sig-name descname">i0</code><span class="sig-paren">(</span><em class="sig-param">x</em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/numpy/numpy/blob/v1.18.1/numpy/lib/function_base.py#L3047-L3107"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numpy.i0" title="Permalink to this definition">¶</a></dt>
<dd><p>Modified Bessel function of the first kind, order 0.</p>
<p>Usually denoted <img class="math" src="../../_images/math/adcd782280d325715f28dea464d7791b13dfe769.svg" alt="I_0"/>.  This function does broadcast, but will <em>not</em>
“up-cast” int dtype arguments unless accompanied by at least one float or
complex dtype argument (see Raises below).</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>x</strong><span class="classifier">array_like, dtype float or complex</span></dt><dd><p>Argument of the Bessel function.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>out</strong><span class="classifier">ndarray, shape = x.shape, dtype = x.dtype</span></dt><dd><p>The modified Bessel function evaluated at each of the elements of <em class="xref py py-obj">x</em>.</p>
</dd>
</dl>
</dd>
<dt class="field-odd">Raises</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>TypeError: array cannot be safely cast to required type</strong></dt><dd><p>If argument consists exclusively of int dtypes.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p><a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.i0.html#scipy.special.i0" title="(in SciPy v1.4.1)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.special.i0</span></code></a>, <a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.iv.html#scipy.special.iv" title="(in SciPy v1.4.1)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.special.iv</span></code></a>, <a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.ive.html#scipy.special.ive" title="(in SciPy v1.4.1)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.special.ive</span></code></a></p>
</div>
<p class="rubric">Notes</p>
<p>The scipy implementation is recommended over this function: it is a
proper ufunc written in C, and more than an order of magnitude faster.</p>
<p>We use the algorithm published by Clenshaw <a class="reference internal" href="#rfd38a370b188-1" id="id1">[1]</a> and referenced by
Abramowitz and Stegun <a class="reference internal" href="#rfd38a370b188-2" id="id2">[2]</a>, for which the function domain is
partitioned into the two intervals [0,8] and (8,inf), and Chebyshev
polynomial expansions are employed in each interval. Relative error on
the domain [0,30] using IEEE arithmetic is documented <a class="reference internal" href="#rfd38a370b188-3" id="id3">[3]</a> as having a
peak of 5.8e-16 with an rms of 1.4e-16 (n = 30000).</p>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="rfd38a370b188-1"><span class="brackets"><a class="fn-backref" href="#id1">1</a></span></dt>
<dd><p>C. W. Clenshaw, “Chebyshev series for mathematical functions”, in
<em>National Physical Laboratory Mathematical Tables</em>, vol. 5, London:
Her Majesty’s Stationery Office, 1962.</p>
</dd>
<dt class="label" id="rfd38a370b188-2"><span class="brackets"><a class="fn-backref" href="#id2">2</a></span></dt>
<dd><p>M. Abramowitz and I. A. Stegun, <em>Handbook of Mathematical
Functions</em>, 10th printing, New York: Dover, 1964, pp. 379.
<a class="reference external" href="http://www.math.sfu.ca/~cbm/aands/page_379.htm">http://www.math.sfu.ca/~cbm/aands/page_379.htm</a></p>
</dd>
<dt class="label" id="rfd38a370b188-3"><span class="brackets"><a class="fn-backref" href="#id3">3</a></span></dt>
<dd><p><a class="reference external" href="http://kobesearch.cpan.org/htdocs/Math-Cephes/Math/Cephes.html">http://kobesearch.cpan.org/htdocs/Math-Cephes/Math/Cephes.html</a></p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">i0</span><span class="p">(</span><span class="mf">0.</span><span class="p">)</span>
<span class="go">array(1.0)  # may vary</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">i0</span><span class="p">([</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span> <span class="o">+</span> <span class="mi">2</span><span class="n">j</span><span class="p">])</span>
<span class="go">array([ 1.00000000+0.j        ,  0.18785373+0.64616944j])  # may vary</span>
</pre></div>
</div>
</dd></dl>

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